Why is Blender moving objects with locked locations on the Y axis after exporting to FBX?
Well, when I imported the FBX into Unity, it was the object called "rig" (the armature object generated by Rigify) that was being moved around. Checking that object for strange translations in each Action, I found nothing.

But I did notice a stray "LocRot" channel that didn't seem to be attached to any particular object. At least not by name. This was what was moving the rig around.
So removing this channel fixed my problem.